Longitudinal bending of a flexible rectilinear rod with an initial imperfection in the form of an initial deflection is considered. The inhomogeneity of the stress state in the bent state leads to constrained deformations causing variability of the elastic characteristics of the material. To obtain the equation of the rod in the deflected state, the theory of deformation of bodies in inhomogeneous stress fields with induced anisotropy of properties proposed earlier by the authors is used. The solution is based on the variational analysis of the rod deflections using the Bubnov-Galerkin method. The growth of the maximum deflection at the steps of successive loading with increasing force is analyzed. Three variants of initial curvatures, including vanishing small curvature, are considered. Approximate integration is performed using simple basis functions. Regardless of the degree of initial curvature and the choice of approximating function, a marked increase in the compressive force corresponding to a significant increase in the deflections is found compared to the bifurcation approach.